LONDON—BP PLC said Thursday that it would lay off about 300 people in the North Sea hub of Aberdeen, Scotland, in the biggest sign yet of the problems besetting the U.K.’s primary oil-producing region.

While the world-wide slide in the price of oil has focused attention on the U.S.’s relatively new shale fields—which are partially responsible for the global oil glut—it is the mature, high-cost fields such as those in the North Sea that seem likely to suffer most. At prices much below $75 a barrel or so, some of the North Sea’s...